A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Author

Vladeta Jovovic, Nov 28 2004

Keywords

• easy
• nonn

Comments

First differs from A018892 at a(30) = 15, A018892(30) = 14.

First differs from A343654 at a(210) = 51, A343654(210) = 52.

Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.

In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

Examples

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A070919, A086222, A086165, A048691, A063647.

Positions of 2's through 5's are A000040, A001248, A030078, A068993.

The version for subsets of {1..n} instead of divisors is A015617.

The version for pairs of divisors is A018892.

The ordered version is A048785.

The strict case is A066620.

The version for strict partitions is A220377.

A version for sets of divisors of any size is A225520.

The version for partitions is A307719 (no 1's: A337563).

The case of distinct parts coprime is A337600 (ordered: A337602).

A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.

A007304 ranks 3-part strict partitions.

A014311 ranks 3-part compositions.

A014612 ranks 3-part partitions.

A051026 counts pairwise indivisible subsets of {1..n}.

A302696 lists Heinz numbers of pairwise coprime partitions.

A337461 counts 3-part pairwise coprime compositions.

Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

Programs

Mathematica

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

PARI

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

Formula

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.